Sorry, i'm sure this is an extremely simple problem, but stuck and extremely confused on how the answer is derived here
I'm fine with getting the deriviate, and using the formula to find the area, but I just can't figure out how
we get sqrt(x/x-1) here
and how multiplying these two, becomes the sqrt(x)
step by step explanation would be amazing!! I appreciate your assistance!
$f(x)$ is the punctual radius of the jar.
$2 \pi f(x)$ is the slice perimeter of the jar.
$\sqrt{1+f'(x)^2}dx$. This is esoteric. Ask your book, this is equal to $\sqrt{1+({dy\over dx})^2}dx=\sqrt{dx^2+dy^2}$. As you see, that is a little hypotenuse, for the length of the slice through the curve $f(x)$.
Hence the surface is the slice perimeter by its length, all sum: $$ \int 2 \pi f(x)\sqrt{1+f'(x)^2}dx $$
so... which is your function??